In this section, we show a subroutine (Algorithm REF ) to generate the approximate left singular matrix, which approximates the left singular matrix \(U_{A}\)  of \(A\) . Firstly, let us recall the FKV algorithm [1]}, which is a Monte-Carlo algorithm. It can find a good low rank approximation of a given matrix \(A\) . Without outputting the full matrix, it outputs a succinct description of the matrix \(\widetilde{V}\)  instead, where \(\widetilde{V}\)  is an approximation to the right singular matrix \(V_{A}\)  of matrix \(A\) , and the corresponding low rank approximation is \(\widetilde{A}:= A \widetilde{V} \widetilde{V}^T\) . Then, according to the definition of the statistical leverage scores of a matrix \(A\) , by a similar argument we can use the FKV algorithm to obtain the approximation left singular vectors \(\widehat{U}\)  of a given matrix \(A\) , and have the corresponding low rank approximation \(\widehat{A} := \widehat{U} \widehat{U}^TA\) .
